Isogeny Graph And Its Application In Cryptography

Elliptic curves play an important role in cryptography,with the development of quantum computers,elliptic curves cryptography has also changed from ECC based on discrete logarithm to post-quantum cryptography based on isogeny.The post-quantum cryptography based on isogeny mainly include:CGL-hash function,SIDH key exchange protocol,CSIDH key exchange protocol.The main math-ematical problems are:computing isogenies,computing endomorphsim rings,and the structure of isogeny graph.With elliptic curves playing an important role in post-quantum cryptography,cryp-tographers have extended the object to high-dimensional Abelian varieties,and there are also hash functions and SIDH key exchange protocols.In this thesis,we study problems related to the structure of the isogeny graphs of supersingular elliptic curves and of abelian varieties,and decide the loops and neighbors of certain vertices.In Chapter 1,we first review problems and known results about the supersingular elliptic curve isogeny graph and describe the main results of this thesis,then review problems and known results about the abelian varieties graph and describe our results in this subject.In Chapter 2,we give definitions and properties of elliptic curves,in particular,the definitions and determining criterion of ordinary and supersingular elliptic curves.In Chapter 3,we introduce basic knowledge of algebraic geometry,including algebraic curves,algebraic surfaces and abelian varieties,and also give the definitions and prop-erties of the supersingular and superspecial abelian varieties.In Chapter 4,we introduce the supersingular elliptic curve isogeny graphs and describe their properties,and then introduce the abelian varieties isogeny graphs and study their properties.In Chapter 5,we give an introduction of applications of the supersingular elliptic curve isogeny graph and the abelian varieties isogeny graph in cryptography.In Chapter 6,we prove our main result about the loops and neighbors of the two vertices whose j-invariants are 0 and 1728 in the supersingular elliptic curve isogeny graph.Our proof is based on the structure of the endomorphism ring and Deuring’s cor-respondence theorem.Our problem is then transformed into solving certain Diophantine equations and distinguishing left ideal classes in a maximal order in certain quaternion algebra.We also describe our result about loops and neighbors of the Fp-vertices in the graph.In Chapter 7,we determine the number of loops of the vertices E1728 × E1728 and E0×E0 on the superspecial abelian varieties isogeny graph.Our proof is based on the study of the structure of endomorphism rings of abelian varieties,theory of matrices over non-commutative rings and Diophantine analysis.In Chapter 8,we give prospect of isogeny graph and its application in cryptography.